Continuum Mechanics and Thermodynamics in the Hamilton and the Godunov-type Formulations

TL;DR

This paper reformulates continuum mechanics, including dislocations, heat conduction, mass transfer, and electromagnetic fields, into Hamiltonian and Godunov-type systems, ensuring thermodynamic consistency and mathematical rigor for better analysis and discretization.

Contribution

It introduces a unified Hamiltonian and Godunov-type formulation of complex continuum mechanics models, ensuring thermodynamic compatibility and mathematical well-posedness.

Findings

Mathematically expresses thermodynamic compatibility as degeneracy and conservation laws.

Provides a symmetric hyperbolic PDE system suitable for numerical discretization.

Ensures well-posedness of the initial value problem for complex continuum models.

Abstract

Continuum mechanics with dislocations, with the Cattaneo type heat conduction, with mass transfer, and with electromagnetic fields is put into the Hamiltonian form and into the form of the Godunov type system of the first order, symmetric hyperbolic partial differential equations (SHTC equations). The compatibility with thermodynamics of the time reversible part of the governing equations is mathematically expressed in the former formulation as degeneracy of the Hamiltonian structure and in the latter formulation as the existence of a companion conservation law. In both formulations the time irreversible part represents gradient dynamics. The Godunov type formulation brings the mathematical rigor (the well-posedness of the Cauchy initial value problem) and the possibility to discretize while keeping the physical content of the governing equations (the Godunov finite volume discretization).